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Flat deformation theorem and symmetries in spa etime.

Josep Llosa

1, Jaume Carot

2

1Departament de Físi a Fonamental, Universitat de Bar elona, Spain

2Departament de Físi a, Universitat de les Illes Balears, Spain

O tober 26, 2018

Abstra t

The at deformation theorem states that given a semi-Riemannian analyti metri g on

a manifold, lo ally there always exists a two-form F , a s alar fun tion c, and an arbitrarily

pres ribed s alar onstraint depending on the point x of the manifold and on F and c, say

Ψ(c, F, x) = 0, su h that the deformed metri η = cg − ǫF 2is semi-Riemannian and at. In

this paper we rst show that the above result implies that every (Lorentzian analyti ) metri

g may be written in the extended Kerr-S hild form, namely ηab := a gab − 2b k(alb) where η is

at and ka, la are two null ove tors su h that kala = −1; next we show how the symmetries

of g are onne ted to those of η, more pre isely; we show that if the original metri g admits

a Conformal Killing ve tor (in luding Killing ve tors and homotheties), then the deformation

may be arried out in a way su h that the at deformed metri η `inherits' that symmetry.

1 Introdu tion

It has been re ently proved [1 that, given a semi-Riemannian analyti metri gab on a manifold M,

there exists a 2-form Fab and a s alar fun tion c su h that:

1. An arbitrary s alar onstraint Ψ(c, Fab, x) = 0, x ∈ M, is fullled and

2. The so- alled `deformed metri '

ηab = cgab − ǫF 2ab where ǫ = ±1 and F 2

ab := FacgcdFdb (1)

1

is semi-Riemannian and at

This result was alled Flat Deformation Theorem . For the purposes of the present paper, we shall

only onsider the four-dimensional Lorentzian ase.

The proof of the above theorem was based on the existen e of solutions for a partial dierential

system that is derived from the ondition that ηab is at. As a onsequen e of the arbitrariness in

the hoi e of the Cau hy hypersurfa e and Cau hy data, the deformation (1) leading to a at ηab is

by no means unique. Furthermore, as the Cau hy-Kovalewski theorem is a ornerstone in the proof,

the validity of the theorem is limited to the analyti ategory.

The purpose of the present paper is to deal with the question of how the symmetries of the metri

gab are ree ted upon the deformed metri ηab, more pre isely: assuming that gab admits a Killing

ve tor eld Xa, we ask whether it is possible to hoose Fab and c in (1) su h that Xa

is also a

Killing ve tor eld for ηab. We shall prove that the answer is in the armative in the ase of

non-null Killing ve tors and that the symmetry is thus somehow `inherited' along the deformation.

The paper is stru tured as follows: se tion 2 ontains some algebrai developments on the on-

sequen es of the deformation law (1) for a 4-dimensional spa etime whi h will allow us to state

it in a number of alternative ways, thus illustrating dierent features of the deformation law. In

se tion 3 we present the formalism and prove some intermediate results

1

in order to demonstrate

the theorem alluded to in the previous paragraph. It is worth noti ing that in order to prove it,

the problem is reformulated on the 3-dimensional quotient manifold (see se tion 4.2), so that a

dimensional redu tion o urs. Se tion 5 ontains a generalization of the above result to the ase

of (non-null) Conformal Killing Ve tors. Finally, in se tion 6, we present some examples whi h we

believe may be of interest due to their physi al relevan e. We put some te hni al developments in

the appendi es in order to make the paper more readable. Also for this reason, we do not insist at

every intermediate step on the lo al hara ter of the results presented here, but the reader should

bear this in mind.

1

This formalism was developed in a number of referen es, notably [4 and [5 whi h will be used in se tion 4. We

present it here in a way well suited to our purposes.

2

2 Algebrai onsequen es of the deformation law

Consider now the 2-form Fab whose existen e is granted by the deformation theorem [1; there are

two possibilities, either it is

(a) singular (or null), then, a tetrad xa, ya, ka, la exists su h that gab = xaxb+ yayb− 2k(alb)

and

Fab = 2k[axb] and then F 2ab = −kakb (2)

or else it is

(b) non-singular (or non-null), in whi h ase a tetrad su h as the one above exists in terms of

whi h Fab reads

Fab = −2B x[ayb] + 2E k[alb] and then F 2ab = −B2 (xaxb + yayb)− 2E2 k(alb) (3)

where E and B are fun tions related to the algebrai invariants of F ab := gacFcb. If either

B or E is zero, the resulting 2-form is timelike or spa elike respe tively. If neither of them

vanishes, the 2-form is said to be non-simple.

In the singular ase, the deformation law (1) reads ηab = c gab + ǫ kakb or, equivalently,

gab =1

cηab −

ǫ

ckakb (4)

with kaka = 0 and ηab at. That is, gab is a onformal Kerr-S hild metri [2. The singular ase is

therefore non-generi and en ompasses a rather restri ted lass of metri s.

In the non-singular ase, from equations (1) and (3) we have that:

ηab = a gab + b Sab (5)

with a = c + ǫB2, b = −ǫ (B2 + E2) and Sab = −2 k(alb). As it was shown in [1, this is the

generi ase in the sense that the at deformation (5) an always be a hieved for any analyti

semi-Riemannian metri .

Noti e that the arbitrary s alar onstraint Ψ(c, Fab, x) = 0 has no onsequen es on the fa tors a

and b in (5). Indeed, in luding (3) the s alar onstraint may be written as f(c,E,B) = 0 or,

3

equivalently, as a relation f(c, a, b) = 0 whi h, at most, an be used to determine c = c(a, b) to

hoose one amongst the many 2-forms Fab ompatible with (5).

We have hitherto proved that:

Proposition 1 Let gab be a Lorentzian analyti metri on a spa etime M. Lo ally there exist two

s alars, a and b, and two null ve tors, ka and la, su h that kala = −1 and the metri

ηab := a gab − 2b k(alb) (6)

is Lorentzian and at.

The above expression vaguely reminds a onformal Kerr-S hild transformation, but in the present

ase two non-parallel null ve tors, ka and la, o ur. We shall hen eforth all this expression extended

Kerr-S hild form and proposition 1 an be restated as:

Any Lorentzian analyti metri an be written in extended Kerr-S hild form.

An equivalent statement is

Proposition 2 Let gab be a Lorentzian analyti metri on a spa etime M. Lo ally there exist two

s alars, a and b, and a hyperboli 2-plane Sab su h that the metri

ηab := a gab + b Sab (7)

is Lorentzian and at.

Noti e that Sab is a 2-dimensional proje tor:

SadS

db = Sa

b , Saa = 2 (8)

whi h proje ts ve tors onto the hyperboli plane spanned by ka, la. If we now denote Hab :=

gab − Sab, i.e. the omplementary proje tor, then:

HadH

db = Ha

b , Haa = 2 , and Sa

dHdb = Ha

dSdb = 0 (9)

4

Hab is then the ellipti 2-plane spanned by any two spa elike ve tors orthogonal to Sab, in parti ular

xa, ya, the spa elike ve tors in the hosen tetrad, i.e. Hab = 2x(ayb), and it is then possible to write

the deformation (1) in a way similar to that given by (7) but in terms of the (ellipti ) proje tor Hab

instead of the Sab, namely:

ηab := a gab + b Hab (10)

where a and b are s alars.

From the omments and developments above and taking (7) into a ount, we an write

gab := Hab + Sab and ηab := (a+ b)Sab + aHab , (11)

that is, the almost-produ t stru ture [3 dened by Sab is ompatible with both metri s, gab and

ηab, and therefore we an state

Proposition 3 Let gab be a Lorentzian analyti metri on a spa etime M. Lo ally it exists a

Lorentzian at metri ηab that shares with gab an almost-produ t stru ture.

3 Spa etimes admitting a (non-null) Killing ve tor

In this se tion we are going to set up and develop the formalism and basi results whi h will be

later used in order to prove the result stated in the introdu tion; namely: that if the metri admits

an isometry, it is always possible to preserve it in the at deformed metri .

Let M be a spa etime with an arbitrary metri ηab2

admitting a Killing ve tor Xa. Let ξa := ηabX

b

and l := ξaXa. Assume that the Killing is non-null, that is: l 6= 0, and denote by S the set of all

orbits of Xa, whi h we assume to be a 3-manifold (the quotient manifold)

3

.

We shall designate by π the anoni al proje tion π : M −→ S where π(x) = Ox is the orbit through

the point x ∈ M of the 1-parameter group generated by Xa.

The proje tor:

hab := δab − 1

lXaξb (12)

2

Note: ηab does not designate the at metri at this point. We use this notation here for later onvenien e.

3

It an be shown that lo ally this is always the ase if xed points of Xaare ex luded.

5

proje ts ve tors in TM onto ve tors that are transverse (orthogonal) to Xa. There is a bije tion

[4 between tensor elds T ′a...b... on S and the tensor elds T a...

b... on M that fulll:

XbT a...b... = 0 , ξaT

a...b... = 0 and LXT a...

b... = 0 (13)

that is, those whi h are transverse to Xaand ξa and Lie invariant along Xa

. Following Gero h [4

While it is useful on eptually to have the three-dimensional manifold S, it plays no further logi alrole in the formalism. We shall hereafter drop the primes: we shall ontinue to speak of tensor elds

being on S, merely as a shorthand way of saying that the eld (formally, on M) satises (13)

As l 6= 0 the proje ted metri

hab := ηab −1

lξaξb (14)

indu es a semi-Riemannian metri on the quotient manifold S, the so- alled `quotient metri '. Its

signature is +1 + 1 − sign (l). We shall designate by hab := ηab − 1

lXaXb

the inverse quotient

metri , that is: habhbc = hac .

3.1 The Killing equation

From LXηab = 0 it follows that ∇aξb is skew-symmetri , that is: ∇bξa +∇aξb = 0 where ∇ stands

for the ovariant derivative asso iated to η.

We also have that LXξa = 0 and Xala = 0, where la := ∇al. Sin e Xais non-null, ∇aξb an be

de omposed as:

2∇aξb := 2 f[aξb] +Θab with f := log |l| and ΘabXb = 0 (15)

Θab = −Θba is related with the vorti ity of the Killing ow. We shall use the above form for the

Killing equation in the sequel.

3.2 The Levi-Civita onne tion on S

Let T a...b... be a tensor eld on S and dene:

DcTa...b... := hamhnb h

kc∇kT

m...n... (16)

6

Clearly, it is a tensor eld on S, sin e T a...b... and hab both satisfy (13), and, sin e Xa

is a KV, the Lie

derivative with respe t to it ommutes with ∇; further it an be easily proved that Da is a linear

onne tion: indeed, it is linear, it satises the Leibniz rule and for any s alar fun tion f on S, Daf

is the gradient of f . Moreover, it an be also shown that it is torsion-free and that Dchab = 0 (this

last result holds trivially); therefore, D is the Levi-Civita onne tion on S (see [4).

Let now va, wbbe two ve tor elds on S, then taking into a ount (13), (15) and (16) one easily

gets

Dvwa = ∇vw

a +1

2lXaΘbcv

bwc(17)

where Dvwa := vbDbw

a. Noti e the formal similarity between this formula and Gauss equation for

hypersurfa es, even though S is not a submanifold and we have the skew-symmetri Θbc instead of

the se ond fundamental form.

3.3 The Riemann tensor on S

Consider next a ve tor eld va on S endowed with the quotient metri hab and its asso iated Levi-

Civita onne tion Da as dened above in (16). We aim at al ulating the Riemann tensor Rcdab for

this onne tion.

From the Ri i identities, [Da,Db]vc = vdRc

dab, we have that

Rabcd = R⊥abcd +

1

2l

(

ΘabΘcd +Θ[acΘb]d

)

,

where R⊥abcd := hma hnb h

pch

qdRmnpq. Using the identity ΘabΘcd + ΘacΘdb + ΘadΘbc = 0 that follows

from the fa t that dimS = 3, we then arrive at

Rabcd = R⊥abcd +

3

4lΘabΘcd (18)

The remaining omponents of Rabcd follow from the se ond order Killing equation [6, ∇a∇bξc =

RdabcXd := RXabc whi h, taking into a ount (15), leads to:

R⊥Xabc =

1

2DaΘbc +

1

2f[bΘac] (19)

RXaXc = −1

2Dalc −

1

4Θ b

a Θbc +1

4llalc (20)

7

We have thus shown that the entire Riemann tensor on M may be expressed in terms of the

kinemati invariants of ξa and the Riemann tensor on S asso iated to the Levi-Civita onne tion

Da of the proje ted (quotient) metri hab.

3.4 Lift of a metri from S to M

We have hitherto shown how a semi-Riemannian metri an be proje ted from M to S. We shall

now onsider the onverse ase. As before, let Xabe a ve tor eld on M and let S be the set of its

orbits, whi h we take to be a manifold a ording to the reasoning at the beginning of the present

se tion. Further, let π : M → S be the anoni al proje tion.

Let now hab be a semi-Riemannian metri on S having onstant signature (+ + σ), σ = ±1. We

shall denote by the same symbol the pulled ba k metri onM, i.e.: π∗hab = hab, whi h is degenerate

be ause habXb = 0, moreover, LXhab = 0. The point now is: does it exist a metri ηab on M su h

that: (a) admits Xaas a Killing ve tor and (b) has hab as the quotient metri ?

If it exists, a relation similar to (14) must hold, with ξa := ηabXband l = ξaX

a. Hen e, the solution

is not unique, be ause we may hoose any ove tor ξa su h that LXξa = 0 and that l := ξaXahas

onstant sign

4

; then taking

ηab := hab +1

lξaξb (21)

as the lifted metri , all the required onditions are satised (namely: Xais a KV of ηab and hab

is its quotient metri ). Then, if no further ondition is demanded, equations (18), (19) and (20)

merely relate the Riemann tensors for both metri s, ηab and hab. However, if we require the lifted

metri ηab to fulll some supplementary ondition, e.g. to be at, then these be ome equations on

the hosen ξa and the given hab, mu h in the same way as the Gauss urvature equation and the

Codazzi-Mainardi equations set up onditions on the way that a submanifold an be immersed in

an ambient spa e.

The hoi e of ξa is restri ted by the ondition LXξa = 0. Assume that a 1-form αa on M su h that

αaXa = 1 and LXαa = 0 is given. Then, the sought ξa an be written as ξa = l(αa + µa), with

l := ξaXaand µaX

a = 0. It an be easily proved that:

LXξa = 0 ⇔ Xl = 0 and LXµa = 0

4

The sign is to be hosen so that the lifted metri has the required signature (+ + +−)

8

Hen e, given a 1-form αa on M su h that αaXa = 1 and LXαa = 0, hoosing ξa is equivalent to

hoosing a fun tion l 6= 0 on S, a 1-form µa on S and taking ξa = l(αa + µa).

The exterior derivative of this expression yields

(dξ)ab =2

ll[aξb] + l (dµ)ab and Θab = l (dµ)ab + l (dα)ab (22)

where (15) has been taken into a ount.

In terms of l and µa, taking (22) into a ount, the equations (18), (19) and (20) read:

R⊥abcd = Rabcd −

3l

4(dµ)ab(dµ)cd (23)

R⊥Xabc =

1

2Da [l(dµ)bc] +

1

2l[b(dµ)ac] (24)

RXaXc = −1

2Dalc −

l2

4(dµ)ad(dµ)bch

bd(25)

that are equations for l, µa and hab to be solved on S.

3.5 Hypersurfa es and Killing ve tors

Let Σ be a surfa e in S, then π−1Σ is a hypersurfa e in M and the Killing ve tor Xais tangent to

it. The following diagram is ommutative:

(η,∇, R) M π// S (h,D,R)

(η′,∇′, R′) π−1Σπ

//

J

OO

Σ

j

OO

(h′,D′,R′)

where J and j are the respe tive embeddings.

We respe tively denote by η′ab, Φab, ∇′and R′

abcd the rst and se ond fundamental forms, the indu ed

onne tion and the intrinsi urvature on π−1Σ as a hypersurfa e of the Riemannian manifold

(M, ηab). Similarly, we denote by h′ab, φab, D′and R′

abcd the orresponding obje ts on Σ regarded

as a hypersurfa e in (S, hab).

Let nabe the unit ve tor η-normal to π−1Σ. Sin e Xa

is tangent to π−1Σ, then ξana = 0. Further-

more, LXna = 0. Indeed, for any V atangent to π−1Σ we have that LXV a

is also tangent to π−1Σ

9

and, using that Xais a Killing ve tor eld, we easily arrive at ηabLXnaV b = 0, whi h implies that

LXna ∝ na. On the other hand, as na

is unit, ηabLXnanb = 0, when e it follows that LXna = 0.

Therefore, nais also a ve tor in S and is the unit ve tor h-normal to Σ.

It an be easily proved that the se ond fundamental forms for π−1Σ and Σ satisfy that: φab = Φ⊥ab.

On the other hand, for any ve tor eld V btangent to π−1Σ, we have that

ΦabXaV b = ∇XnbV

b = −∇V ξbnb = −1

2(dξ)abV

anb

where in the se ond equality we have used that LXV ana = 0 and that V bnb = 0. The above

equation implies, putting (dξ)abnb := (dξ)an and fbn

b := fn,

ΦabXa =

1

2(dξ)nb =

1

2fnξb +

1

2Θnb (26)

where (15) has been taken into a ount. Therefore,

Φab = φab +1

2lfn ξaξb +

1

lΘn(aξb) (27)

4 Flat deformation

The aim of this se tion is to prove the main result in this paper, namely,

Theorem 1 Let (M, gab) be a spa etime with a metri gab admitting a non-null Killing ve tor Xa.

Lo ally there exists a deformation law

ηab = a gab + bHab (28)

where a and b are two s alars, Hab is a 2-dimensional proje tor on a g-ellipti plane and ηab is at

and also admits Xaas a Killing ve tor.

It will be onvenient for our purposes to prove the following result previously:

Proposition 4 Let Xabe a Killing ve tor for gab and let ηab be dened by (28) with b 6= 0, then:

LXηab = 0 ⇔ LXa = LXb = 0 and LXHab = 0 (29)

10

Proof: As LXgab = 0, LXηab = 0 implies that

LXa gab + LXbHab + bLXHab = 0 (30)

Sin e Hab is a 2-dimensional proje tor, HabHab = Ha

a = 2, and taking the Lie derivative we get

2LXHabHab = 0. Contra tion of (30) with gab and Hab

leads respe tively to

4LXa+ 2LXb = 0 and 2LXa+ 2LXb = 0

whi h imply: LXa = LXb = 0. Substituting ba k into (30) and taking into a ount that b 6= 0

yields LXHab = 0.

The proof of theorem 1 spreads over the present se tion and it onsists in nding a, b and Hab su h

that:

(i) ηab = agab + bHab is at and

(ii) LXa = LXb = 0 and LXHab = 0.

The number of unknowns is 6, namely: 2 for a and b plus 4 forHab (re all the onstraintsHacH

cb = Ha

b

and Haa = 2). Then, (i) means that the Riemann tensor for ηab vanishes:

Rabcd = 0 (31)

To ensure (ii) we shall solve (31) on S and then pull the solutions ba k to π−1S = M.

We rst introdu e the de ompositions:

gab = pab +1

lξaξb and ηab = hab +

1

lξaξb (32)

where ξa := gabXband l = ξbX

bare known from the data gab and Xb

, whereas

ξa := ηabXb = aξa + bHabX

b and l := ξaXa , (33)

depend on the unknowns. Noti e that bHabXaXb = l − al.

11

4.1 The proje tion of our problem onto the quotient manifold S

We must now repla e the unknowns (a, b, Hab), whi h are tensor quantities on M, with others that

are tensor quantities on S. Consider the ove tor αa = ξa/l. It is obvious that αaXa = 1 and

LXαa = 0; hen e the results in se tion 3.4 an be applied and we have that µa =1

lξa −

1

lξa is a

ove tor in S, thus we an write

ξa = lmνa +l

lξa , (34)

where νa is a p-unitary ove tor on S and m :=√

µaµbpab. Then, on a ount of (33), we have that

5

:

HabXb =

1

b

(

lm νa +l − al

lξa

)

(35)

Now, Hab is a 2-dimensional proje tor and therefore its eigenvalues are 0 and 1, both with multipli ity

2. HabX

bis an eigenve tor (not unit), and a se ond one may be hosen so that it is g-orthogonal to

it. We an thus write:

Hab = βaβb + ωaωb (36)

where βa and ωa are g-unitary and mutually g-orthogonal, and

βa =lm

√

b(l − al)νa +

1

l

√

l − al

bξa (37)

Sin e Hab is a proje tor, it follows that ωaX

a = ωaνa = 0, and as βa is g-unitary we also have that:

l2m2

l − al= b+ a− l

l(38)

From LXHab = 0 (Proposition 4), its transverse proje tion

Hab =(b+ a)l − l

blνaνb + ωaωb (39)

satises also LXHab = 0. Hen e, Hab is a tensor on S.5

We expli itly ex lude the ases l − al = 0 and b = 0 sin e they are non-generi . Note that b = 0 orresponds to

the metri η being onformally at.

12

The quotient metri hab is the transverse proje tion of ηab and, taking (28), (32) and (39) into

a ount, we obtain:

hab = apab +

(

b+ a− l

l

)

νaνb + b ωaωb (40)

We have seen so far that the set of unknowns a, b, Hab tensor quantities on M an be

assigned the new set of unknowns a, b, l, νa, ωb, where νa and ωb are p-unitary and mutually

p-orthogonal ove tors on S, and a, b and l are s alar fun tions on S. The inverse orresponden eis easily established. It su es to take Hab as dened by (36) with β dened by (37)

(Noti e that the number of degrees of freedom is still 6 be ause, on e νa is given, the unit orthogonal

ove tor ωa is determined by only giving one angle.)

Due to the symmetries of the Riemann tensor, Rabcd, it an be separated as:

Rabcd = Labcd +2

l

(

Lab[cξd] + Lcd[aξb])

+4

l2ξ[bLa][cξd] (41)

where Labcd, Labc and Lac are transverse to Xband have the following symmetries:

(a) Labcd has the same symmetries as a Riemann tensor in 3 dimensions,

(b) Labc = −Lbac, Labc + Lbca + Lcab = 0 and Lab = Lba

Noti e that:

Labcd = R⊥abcd , Labc = R⊥

abcX and Lac = RXaXc (42)

and are given by (18), (19) and (20). Then equations (31) atness of ηab are equivalent to:

Labcd = 0 , Labc = 0 and Lac = 0 (43)

By taking the exterior dierential of mνa and taking (34) and (22) into a ount, we have that

2D[a

(

mνb])

=1

lΘab −

1

lΘab (44)

with m given by (38). In luding now (18), (19), (20), (40) and (44), the equations (43) result in

se ond order partial dierential equations relating a, b, l, νa and ωb, i.e. tensor quantities on S.

13

4.2 The onstraints and the redu ed system

Equations (43) onstitute a system of 20 independent equations for only 6 independent unknowns.

To handle this overdetermination we shall take 6 equations among them as a redu ed partial dier-

ential system [7, that we shall solve by giving Cau hy data on a non- hara teristi surfa e Σ [8.

The remaining 14 equations are to be onsidered as onstraints to be fullled by the Cau hy data

on Σ. It must be then proved that any given solution of the redu ed PDS fullling the onstraints

on Σ also fullls them on a neigbourhood of Σ.

Given a surfa e Σ ⊂ S, we hoose Gaussian p-normal oordinates (x1, x2, x3) on a neigbourhood

U ⊂ S of Σ:

x1 = 0 on Σ , p11 = s = ±1 and p1j = 0 , j = 2, 3 (45)

The sign s depends on the sign of l: if l < 0, then s = +1, while for l > 0, s an take both values ±1.

For the sake of simpli ity, here we shall hoose Σ so that s = − sign (l) and then pij has signature

(++).

In these oordinates, we hoose (indi es a, b, c, . . . run from 1 to 3 and i, j, . . . run from 2 to 3)

L11 = 0 , L1j1 = 0 , L1i1j = 0 (46)

as the redu ed partial dierential system and

Laj = 0 , Lbijk = 0 , Ljcd = 0 (47)

as the onstraints. (Noti e that L1jk = 0 is in luded in the above equalities be ause, as a onsequen e

of the rst Bian hi identity, L1jk = −Ljk1 − Lk1j .)

In Appendix A we prove that, if a, b, Hab is an analyti solution of the redu ed PDS (46) fullling

the onstraints (47) on Σ, then the onstraints are also fullled in an open neigbourhood of Σ.

4.3 The redu ed PDS

We shall now write equations (46) in terms of the unknowns a, b, l, νa, ωa. We shall only make

expli it the prin ipal parts, i.e. those terms involving se ond order partial derivatives with respe t

to the oordinate x1. In what follows a dot will stand for ∂1, whereas ∼= will mean equal apart

from non-prin ipal terms.

14

(a) From (42) and (20), and taking into a ount that l 6= 0, we have that L11 = 0 leads to

l ∼= 0 . (48)

(b) From (42) and (19), in luding (44), we obtain Labc∼= −Dc

[

l D[a(mνb])]

. Therefore, L1j1 = 0

amounts to

mνj +mνj ∼= 0 , j = 2, 3 (49)

with m given by (38).

( ) From (42) and (18) we have that the third of the equations (46) L1i1j = 0 leads to

hij ∼= 0 (50)

whi h, using (39), (40), (48) and (49), be omes

a(

pij + [l(b+ a)− l]νiνj)

+ b ωiωj + b [ωiωj + ωiωj ] ∼= 0 , i, j = 2, 3 (51)

The hara teristi determinant for the redu ed partial dierential system onstituted by the

six equations (48), (49) and (51) is (see Appendix B for details):

∆ := 2bω21ν1τ1 p

[

1− sω21 + [l(b+ a)− l]ν21

] l − al

l2[(

b+ a− l

l

)

(1− sν21)− sω21

(

a− l

l

)]

4.4 Geometri al meaning of the onstraints

It remains to be shown that Cau hy data fullling the onstraints (47) on the Cau hy surfa e Σ do

exist. Consider π−1Σ, whi h is a hypersurfa e in M, and take oordinates (x1, . . . x4) adapted to

both Xaand π−1Σ, i.e. Xa = δa4 and x1 = 0 on π−1Σ.

Let Pab and P a

b be the proje tors:

Pab := δab − 1

g11g1a δ1b and P a

b := δab −1

η11η1a δ1b (52)

They both proje t ve tors in TM onto the hyperplane T (π−1Σ) and, while Pab proje ts parallelly

to g1a, P ab does it parallelly to η1a. It is obvious that P

1b = P 1

b = 0, hen e

PabP

bc = P a

c and P ab P

bc = P

ac (53)

15

whi h implies that, when restri ted to the hyperplane T (π−1Σ), both proje tors, Pab and P a

b , yield

the identity.

It is easy to see that the onstraints (47) amount to

Rabcd = 0 whenever at most one of the indi es is 1

that is, RabcdPbeP

cfP

dg = 0 whi h, in luding (53) is equivalent to

RabcdPbeP

cfP

dg = 0 (54)

Then, if nais the unit ve tor η-normal to π−1Σ, (54) is equivalent to

Rtangabcd = 0 and Rtang

nbcd = 0 (55)

where tang denotes omponents tangential to π−1Σ and Rnbcd := Rabcdna.

π−1Σ an be seen both as a hypersurfa e of the Riemannian manifold (M, ηab) and as a hypersurfa e

of (M, gab). We shall denote η′ab and g′ab the respe tive rst fundamental forms. The two normal

ve tors are respe tively:

na =1

√

|η11|η1a , na =

1√

|η11|δ1a and na =

1√

|g11|g1a , na =

1√

|g11|δ1a (56)

and the se ond fundamental forms are:

Φab = P ca∇cnb and Φab = P

ca∇cnb

The Gauss urvature equation for π−1Σ as a submanifold of (M, ηab) reads [9:

Rtangabcd = R′

abcd + 2Φa[dΦc]b (57)

and the Codazzi-Mainardi equation is

Rtangnbcd = 2∇′

[dΦc]b (58)

where ∇′and R′

abcd are respe tively the indu ed onne tion and the intrinsi urvature.

16

The onstraints (55) are therefore equivalent to

R′abcd + 2Φa[dΦc]b = 0 and ∇′

[dΦc]b = 0

a parti ular solution of whi h is

Φab = 0 and R′abcd = 0 (59)

The normal derivatives of the unknowns. The rst of the equations (59) determines the rst

order normal derivatives of the unknowns on the Cau hy hypersurfa e Σ. Indeed, from (27) and

Φab = 0 we have that:

fn = 0 , Θnb = 0 , φab = 0 (60)

Furthermore, as Θab is skewsymmetri and φabnb = 0, it is obvious that φab = 0 and Θna = 0 are

equivalent to

φij = 0 and Θnj = 0 , i, j = 2, 3

Noti e that the remaining equations, namely φa4 = 0 and Θn4 = 0, are identi ally satised be ause

φab and Θab are tensors on S and in these oordinates Xa = δa4 .

In luding then (27), (33) and (44), equations (60) are equivalent to:

nbDbl = 0 , 2l D[b

(

mνj])

nb +l

lΘnj = 0 , Dinj = 0 (61)

and, using (56), we have that

Danb =

√

|g11||η11|

(

φab +1

2nbDa log

[ |g11||η11|

]

− bcabnc

)

where bcab is the dieren e tensor for the onne tions D and D.

In Gaussian p-normal oordinates, taking into a ount (40) and writing expli itly the prin ipal

terms only, (61) be omes:

h11 l ∼= 0 , h11 (mνj +mνj) ∼= 0 , h11hij ∼= 0 (62)

The similitude of these equations with (48), (49) and (50) is apparent and the hara teristi de-

terminant is (h11)6∆. Hen e, provided that the Cau hy data on Σ are hosen so that ∆ 6= 0 and

h11 6= 0, the onstraints Φab = 0 permit to obtain the rst order normal derivatives of the unknowns,

namely a, b, l, νb and ωc on Σ, in terms of the values of a, b, l, νb and ωc on Σ.

17

The unknowns on the Cau hy surfa e Σ. The se ond of the equations in (59) is a ondition

on the values of the unknowns on Σ. The isometry group G generated by Xaa ts also on π−1Σ

and π−1Σ/G = Σ. Hen e, relations similar to (18-20) hold:

R′⊥abcd = R′

abcd −1

2l

(

Θ′abΘ

′cd +Θ′

[acΘ′b]d

)

= 0 (63)

R′⊥Xabc =

1

2D′

aΘ′bc +

1

2f ′[bΘ

′ac] = 0 (64)

R′XaXc = −1

2D′

al′c −

1

4Θ′ b

a Θ′bc = 0 (65)

with R′ := J∗R , R′ := j∗R , Θ′ := j∗Θ , l′ = j∗l.

As Σ has only two dimensions, Θ′acΘ

′ bc = θ′ 2h′ ba , where 2θ′ 2 = Θ′bcΘ

′ bc. Hen e, equation (64)

is equivalent to Θ′ bcR′⊥Xabc = 0 whi h, after a little algebra yields D′

aθ′ 2 + f ′

aθ′ 2 = 0 and, sin e

f ′ = log |l′|, we have that:θ′ 2l′ = constant (66)

In two dimensions, the Riemann tensor has only one independent omponent: R′abcd = R′ h′a[ch

′d]b ,

therefore (63) and (65) are respe tively equivalent to

R′ =3θ′ 2

2l′and D′

aD′cl′ =

1

2θ′ 2 h′ac (67)

The integrability onditions for this equation imply that θ′ = 0. Indeed, as

D′bD

′aD

′cl′ −D′

aD′bD

′cl′ = −R′d

cbaD′dl

′ ,

we have that D′[bθ

′ 2h′a]c = −R′D′[bl

′h′a]c , where the fa t that we are in 2 dimensions has been

used to simplify the Riemann tensor. Taking now into a ount the rst equation (67) we obtain

D′bθ

′ 2 − 3θ′ 2

2l′ D′bl′, or θ′ 2/l′ 3 = onstant or θ′ 2l′−3 = onstant. This, together with (66) implies

l′ = onstant whi h substituted in (67) leads to θ′ = 0.

Therefore, equations (63-65) are equivalent to:

R′ = 0 , θ′ = 0 and D′aD

′cl′ = 0 (68)

The Gaussian p-normal oordinates introdu ed in se tion 4.2, equation (45), are spe ially well suited

to our problem. In these oordinates ve tors that are tangent to Σ are hara terized by v1 = 0

18

and the restri tion to Σ of any ovariant tensor on S, Tab... , a, b, . . . = 1, 2, 3 , merely onsists

in keeping the omponents Tij... , i, j, . . . = 2, 3 . Thus, h′ij := (j∗h)ij = hij , ν′i := (j∗ν)i = νi ,

Θ′ij := (j∗Θ)ij = Θij , m

′ := m j = m and so on.

Now, in luding this and the se ond equation (68), the restri tion to Σ of equation(44) is

2D′[i

(

mνj])

= −1

lΘij , i, j = 2, 3 (69)

and, as all dierential forms in Λ2Σ are losed, this equation is lo ally integrable and yields mνj ,

j = 2, 3.

Moreover, l′ = onstant is a solution of the third equation (68) and therefore we shall take l = onstant

on Σ.

As Σ has only two dimensions, R′ = 2ǫ′ ij(h′)ǫ′ kl(h′)R′ijkl, where ǫ′ ij(h′) is the volume tensor on Σ

for the metri h′kl. In two dimensions the volume tensors ǫ′ ij(h′) and ǫ′ ij(p′) are proportional to

ea h other and therefore R′ = 0 is equivalent to ǫ′ ij(p′)ǫ′ kl(p′)R′ijkl = 0 , or

pikpjlR′ijkl = 0 (70)

This is a ondition on h′ij whi h depends on the unknowns a, b, l, νa, ωb, a, b = 1, 2, 3 .

From the third equation (68) we know that l = onstant on Σ. Then, by solving equation (69) we

obtainmνj , j = 2, 3, on Σ. We then hoose ωi, i = 2, 3, on Σ whi h, together with the orthogonality

onditions

ωaωbpab = νaνbp

ab = 1 and νaωbpab = 0 ,

permit to obtain ωb, νa, a, b = 1, 2, 3 and m. Finally, substituting this in (38), we an obtain

b = b(a) and therefore ondition (70) yields a partial dierential equation for a, whose prin ipal

part is

(

νjνk − [1 + pilνiνl] pjk)

∂jka ∼= 0 where νj := pjkνk (71)

The hara teristi form is:

χ(zl) = (zlνl)2 − [1 + pilνiνl] p

jkzjzk

and the existen e of non- hara teristi lines for equation (70) is obvious.

19

4.5 Summary of the proof

So far, we have analyzed the existen e of a solution to the problem stated in se tion 1. Let us now

synthesize a way to nd su h a solution:

(a) From the given Xaand gab, obtain l, ξa, Θab and the quotient metri pab;

(b) Choose a Cau hy surfa e Σj→ S and a hart of Gaussian p-normal oordinates for Σ,

(x1, x2, x3);

( ) Choose mνi, i = 2, 3, on Σ as a solution of 2l∂[i(

mνj])

= −Θij ;

(d) Take l = onstant on Σ;

(e) Then hoose ωi, i = 2, 3, su h that inequality ∆ 6= 0 is fullled and, in luding the orthonor-

mality ondition, the denition (33) and the obtained value for mνj , derive ω1, ν1 and m on

Σ;

(f) With the relation (38) obtain b = b(a) and

(g) Substitute the above in (70) and solve it to obtain a on Σ.

With this we have a, b, l, νc, ωd on Σ. Then

(g) Solve (61) to derive a, b, l, νc, ωd on Σ; and

(h) With these Cau hy data, solve the redu ed partial dierential system (46); then use (34) to

have ξa, (40) to have hab and (32) to have ηab.

5 Generalization to Conformal Killing Ve tors

The main result in this paper, stated in theorem 1, an be extended almost immediately to the ase

of Conformal Killing Ve tors (CKV for short), as a onsequen e of the so alled Defrise-Carter's

theorem (see for instan e [11); whi h states, roughly speaking, that given a (non- onformally at)

metri g admitting an r-dimensional Lie algebra of CKVs, Cr , there exists a fun tion Ω, su h that

Cr be omes a Lie algebra of Killing ve tors for the onformally related metri g = Ω2g.

20

Thus, we an state:

Theorem 2 Let (M, g) be a spa etime su h that the metri gab admits a non-null CKV Xa. Lo ally,

there exists a deformation law as the one given by (28) su h that Xais a KV for the at metri ηab.

Proof: Sin e Xais a CKV of the metri gab, there exists a onformal fa tor Ω2

su h that gab :=

Ω2gab has Xaas a KV [11. By theorem 1, it then follows that a at, deformed metri ηab exists,

ηab = agab + bHab

for whi h Xais a KV, dening next a := Ω2a and taking into a ount the above expression for ηab

as well as the relation between the metri s g and g, it readily follows that Xais a KV of the at

metri

ηab = agab + bHab.

6 Examples

We next present some physi ally signi ant examples. We have hosen families of well hara terized

spa etimes and then sele ted, amongst all spa etimes in the family, one well-known and physi ally

relevant parti ular solution. For the sake of onvenien e, instead of the deformation law (28) in

theorem 1, we shall rather use the equivalent formula (7) with the hyperboli proje tor Sab.

6.1 Class A1 warped spa etimes

For these spa etimes, oordinates xa = u, xk with k = 1, 2, 3 exist su h that the metri takes the

following form (see [12 for denitions and further details),

ds2 = ǫdu2 + f2(u)hij(xk)dxidxj , ǫ = ±1

where f is some fun tion of u. For ǫ = +1, u is a spa elike oordinate ( lass A1 spa elike warped),

whereas for ǫ = −1, u is time ( lass A1 timelike warped). In what follows, we shall onsider only

21

the latter ase and put u := t, thus, we shall take the line element to be

ds2 = −dt2 + f2(t)hij(xk)dxidxj, i, j, k = 1, . . . , 3. (72)

Writing now

ds2 = f2(τ)ds2, with dτ =dt

f(t), f(τ) = f(t(τ)) (73)

we get, in an obvious notation,

ds2 = −dτ2 + pij(xk)dxidxj , or else gab = f2(τ)gab. (74)

Now, ∂τ is a KV of gab and a CKV of the original metri gab; further, it is orthogonally transitive.

hen e, pij(xk) is a Riemannian metri on the quotient manifold oordinated by xk, k = 1, 2, 3.

Making use of the equivalent to the at deformation theorem in three dimensions for a Riemannian

metri (see [13), we an see that a s alar fun tion a(xk) and a ovariant ve tor eld µi(xk) exist su h

that they fulll a previously hosen arbitrary relation, say Ψ(a, ||µ||) = 0, where ||µ||2 = pijµiµj ,

with pijpjk = δik, and the metri

ηij = apij + µiµj (75)

is at. Presently, we hoose

Ψ(a, ||µ||) = ||µ||2 + a− 1 = 0,

and it then follows that the 4-dimensional semi-Riemannian metri

η := −dτ ⊗ dτ + ηij(xk)dxi ⊗ dxj

is also at and admits the KV ∂τ .

Using now (75) we have that

η := −dτ ⊗ dτ + apijdxi ⊗ dxj + µidx

i ⊗ µjdxj ,

or else, using the oordinates xa = x1, x2, x3, x4 = τ , setting µ4 = 0 and making use of (73), it turns

out that we an write

ηab = agab − (1− a)δ4aδ4b + µaµb = agab + (1− a)Sab, (76)

22

where

Sab := −δ4aδ4b + µaµb, µa :=

1

||µ||µa

is a two-dimensional hyperboli proje tor (re all that we hose ||µ||2 = 1 − a), and thus (76)

orresponds the sought for form (7).

6.2 Spa etimes with additional symmetries

In some ases with additional symmetries it is possible to derive an expli it expression for µi; this

giving for granted that the deformed metri ηab, the fa tors a and b, and the hyperboli proje tor

Sab will share the same additional symmetries. (Noti e that this is only a onje ture that goes

beyond what has been proved so far, although theorem 1 supports its plausibility.)

As an example, take a stati spheri ally symmetri metri

g = −f2(r)dt⊗ dt+ p2(r)dr ⊗ dr + r2(

dθ ⊗ dθ + sin2 θdφ⊗ dφ)

. (77)

whi h, besides the three KV implementing the spheri al symmetry, admits one fourth KV, namely

∂t. The quotient spa e S an be given the stru ture of a manifold as dis ussed previously. Consider

next the metri h on S,

h = g + f2(r)dt⊗ dt = p2(r)dr ⊗ dr + r2(

dθ ⊗ dθ + sin2 θdφ⊗ dφ)

. (78)

By the theorem in [13 regarding three-dimensional metri s, a s alar a and a ovariant ve tor

µi exist, whi h fulll an arbitrary, previously hosen onstraint, that we shall take Ψ(a, ||µ||) :=

‖µ‖2 − f−2(r) + a = 0, and are su h that the deformed three-dimensional Riemannian metri

η = ah+ µ⊗ µ (79)

is at. Let us next make a guess at a and µ and take a = a(r) and µ = µ(r)dr, we shall have:

||µ||2 = hijµiµj = p−2(r)µ2(r),

hen e

η = (a+ ||µ||2) p2(r)dr ⊗ dr + ar2(

dθ ⊗ dθ + sin2 θdφ⊗ dφ)

. (80)

The spa etime metri η := −dt⊗ dt+ η is also at, i.e.:

η = −dt⊗ dt+ ah+ µ⊗ µ = a(

g + f2(r) dt⊗ dt)

+ µ⊗ µ− dt⊗ dt, (81)

23

whi h is already in the desired form (7) with bS := µ⊗ µ−(

f−2(r)− a)

f2(r) dt⊗ dt .

In order to ensure that S is a hyperboli proje tor as required, we need ‖µ‖2 = f−2(r) − a whi h

is fullled thanks to the hosen arbitrary onstraint Ψ(a, ||µ||) = 0.

Substituting the above ba k into (80) we get that

η = f−2(r)p2(r)dr ⊗ dr + ar2(

dθ ⊗ dθ + sin2 θdφ⊗ dφ)

(82)

must be at, and this determines a up to a onstant. Noti e that a line element of the form

dσ2 = F 2(r)dr2 + Y 2(r)dΩ2

is at i

dY (r)

dr= ±F (r) ,

hen e, hoosing the plus sign for onvenien e and sin e Y 2 = ar2 and F (r) = p(r)/f(r), we nally

get

√a =

1

r

(∫ r p(r′)

f(r′)dr′ +K

)

, K = constant. and µ = p(r)√

f−2(r)− a (83)

Two interesting parti ular ases are the following:

Friedmann-Robertson-Walker spa etimes These are parti ular instan es of the the ones just

dis ussed, namely: lass A1 timelike warped. As it is well known, the metri may be written

as

ds2 = −dt2 +R2(t)

1 + k4r

2

(

dr2 + r2dΩ2)

, dΩ2 = dθ2 + sin2 θdφ2. (84)

Pro eeding as in the general ase in se tion 6.1, we an write ds2 = R2(t)ds2, where

ds2 := −dτ2 +

(

1 +k

4r2)−1

(

dr2 + r2dΩ2)

and dτ :=dt

R(t), (85)

with ∂τ being a KV of the metri g (of line element ds2) and a CKV of g (line element ds2).

The metri g is a parti ular ase of (77) with

f(r) := 1 and p(r) :=

(

1 +k

4r2)−1/2

24

whi h substituted in (83) yield

µ =1− a

√

1 + kr2/4

and

√a =

1

r

(

K +

∫ r

dr′[

1 +k

4r′2]−1/2

)

S hwarzs hild solution Consider next the well-known S hwarzs hild solution written in the form

g = −(

1− rsr

)

dt⊗ dt+(

1− rsr

)−1dr ⊗ dr + r2

(

dθ ⊗ dθ + sin2 θdφ⊗ dφ)

. (86)

whi h is a parti ular ase of (77) with

f(r) :=

√

1− rsr

and p(r) := 1/f(r)

whi h substituted in (83) yield

µ =

√

(1− rs/r)− a (1− rs/r)2

and

√a = 1 +

rsr

[

K + ln

(

r

rs− 1

)]

, K = constant.

A knowledgements

The authors are grateful to B. Coll, A. Molina and J.M. Pozo for interesting omments and sugges-

tions upon reading a previous version of the manus ript. J.C. a knowledges nan ial support from

the Spanish Ministerio de Edu a ión through grant No. FPA-2007-60220. Partial nan ial support

from the Govern de les Illes Balears is also a knowledged. J.Ll. a knowledges nan ial support from

Ministerio de Edu a ión through grant No. FIS2007-63034 and from the Generalitat de Catalunya,

2001SGR-00061 (DURSI). Both authors are also grateful to an anonymous referee for his omments

and suggestions, whi h have greatly ontributed to making the paper more pre ise and readable.

25

Appendix A:

We here prove that the onstraints (47) propagate out of Σ. Assume that a, b and Hab is a solution

of the redu ed PDS (46) for a set of Cau hy data fullling the onstraints (47) on the Cau hy

surfa e Σ. We must prove that these onstraints also hold on a neighbourhood of Σ.

Given a, b and Hab, onsider the metri ηab = agab + bHab. Let ∇ and Rabcd respe tively denote

the Levi-Civita onne tion and the Riemann tensor for ηab. By the se ond Bian hi identity we have

that:

∑

cde

∇eRabcd := ∇eRabcd +∇cRabde +∇dRabec ≡ 0 (87)

In luding (41), the dierent proje tions of this equation are

(a) the proje tion on Xbis:

∑

cde

(

DeLcda −1

2feLcda +

1

lLacΘed −

1

2LabcdΘ

be

)

≡ 0 (88)

(b) the totally transverse proje tion yields:

∑

cde

(

DeLabcd +1

lLabcΘed +

1

lLcd[aΘeb]

)

≡ 0 (89)

( ) and the proje tion on Xeis:

∇XRabcd + 2∇[cRabd]X − 2Rab[de∇c]Xe = 0 (90)

whi h is transverse to X for the indi es c and d.

As Xais a Killing ve tor, LXRabcd = 0, and the above equation be omes

∇[cRabd]X −Re[bcd∇a]Xe = 0 (91)

whi h, proje ted on Xband in luding (41), yields

D[cLad] −1

2Lab[dΘ

bc] −

1

2f[cLad] −

1

4LcdbΘ

ba − l

4Lbacdf

b = 0 (92)

On its turn, the totally transverse omponent of (91) is:

D[cLabd] −1

2Le[bcdΘ

ea] +

1

2Lcd[bfa] +

1

lΘ[c[bLa]d] = 0 (93)

26

In Gaussian normal oordinates equations (88), (89), (92) and (93) respe tively read:

∂1Ljka + 2∂[jLk]1a = lin , ∂1Labjk + 2∂[jLabk]1 = lin

∂1Laj − ∂jLa1 = lin , ∂1∂1Labj − ∂jLab1 = lin

(94)

where j = 2, 3 and a, b, . . . = 1, 2, 3, and lin denotes linear terms not ontaining partial deriva-

tives. (We have only kept those equations governing the propagation outwards of Σ, i. e. those

ontaining partial derivatives with respe t to x1.)

As the metri ηab is a solution of the redu ed PDS (46), we have that L11 = 0 , L1j1 = 0 and

L1i1j = 0. Equations (94) thus yield the following linear partial dierential system to be fullled

by the onstraints (47):

∂1Ljkl = lin + 2∂[jLk]l1 , ∂1Ljk1 = lin

∂1Llijk = lin + 2∂[jLlik]1 , ∂1L1ijk = lin and

∂1L1j = lin , ∂1Lij = lin + ∂jLi1

whi h is already in the normal form for the Cau hy-Kowalevski theorem [8. As the hosen solution

a, b and Hab of (46) is assumed to be analyti , the oe ients are analyti . Then, for the Cau hy

data Laj = 0 , Lbijk = 0 and Ljcd = 0 on Σ, the solution is unique in the analyti ategory and, by

linearity, Laj = 0 , Lbijk = 0 and Ljcd = 0 on an open neighbourhood of Σ.

Appendix B: The hara teristi determinant

The redu ed PDS is onstituted by the six equations (48), (49) and (51):

l ∼= 0 (95)

mνj +mνj ∼= 0 , j = 2, 3 (96)

a(

pij + [l(b+ a)− l]νiνj)

+ b ωiωj + b [ωiωj + ωiωj] ∼= 0 , i, j = 2, 3 (97)

where

m =m

2

(

b+ a

b+ a− l/l+

a

a− l/l

)

as it easily follows from (38) and (95).

27

The surfa e Σ is non- hara teristi if the PDS an be solved for the se ond partial derivatives of

the unknowns, namely a, b, l, νa and ωb on Σ, where a double dot stands for ∂21 . Noti e that due

to the onstraints of p-unitarity and p-orthogonality, in νa and ωb there are only three independent

unknowns. In order to handle them more appropriately we shall onsider the p-orthonormal triad

of spatial ove tors

ωa , νa , τa where τa := ǫabcωbνc,

where ǫabc := ǫabcdXd/l is the p-volume tensor on S.

We then have that:

ωa = Ω3νa − Ω2τa , νa = −Ω3ωa +Ω1τa , τa = Ω2ωa − Ω1νa

and, deriving again and keeping only prin ipal terms:

ωa = Ω3νa − Ω2τa , νa = −Ω3ωa + Ω1τa , τa = Ω2ωa − Ω1νa (98)

whi h introdu ed in (96) and (97) yields

mνj −mωjΩ3 +mτjΩ1∼= 0 , j = 2, 3 (99)

a(

pij + [l(b+ a)− l]νiνj)

+ b ωiωj + 2bν(iωj)Ω3 − 2bτ(iωj)Ω2∼= 0 , i, j = 2, 3 (100)

This last expression (100) ontains three independent equations, whi h amount to the ontra tions

with pij , ωiωj − pijωlωland νiνj − pijνlνl. They read, respe tively:

(

2 + [l(b+ a)− l]νlνl)

a+ ωlωl b+ 2bνjωj Ω3 − 2bτ jωj Ω2∼= 0

[

−ωlωl + [l(b+ a)− l](

(νlωl)2 − νlν

l ωjωj

)]

a ∼= 0

−νlνl a+

(

(νlωl)2 − νlν

l ωjωj

)

b− 2b(

νlωl νjτ

j − νlνl τ jωj

)

Ω2∼= 0

(101)

On its turn, the expression (99) onsists of two independent equations. They are equivalent to the

wedge produ ts with τi and νi, namely

−m (ν ∧ ω) Ω3 +m (ν ∧ τ) Ω1∼= 0

(τ ∧ ν)m

2

(

b+ a

b+ a− l/l+

a

a− l/l

)

−m (τ ∧ ω) Ω3∼= 0

(102)

where (38) has been used and (ν ∧ ω) := ν2ω3 − ν3ω2 and so on.

28

Some simpli ation is gained taking into a ount that ωa, νb, τc is a p-orthonormal triad and, in

the Gaussian p-normal oordinates of se tion 4.2, we have that:

ω ∧ ν = sτ1√p , ν ∧ τ = sω1

√p , τ ∧ ω = sν1

√p

where p := det(pij), and

(ωlνl)2 − ωlωlν

jνj = −1

p(ν ∧ ω)2 = −ν21 , ωlνlν

jτj − ωlτlνjνj = ω1τ1

Furthermore,

ωlωl = 1− sω21 , νjτj = −sν1τ1 , ωjτj = −sω1τ1

Substituting this into (95), (101) and (102), we obtain

(

2 + [l(b+ a)− l](1− sν21))

a+ (1− sω21) b− 2sbν1ω1 Ω3 + 2sbτ1ω1 Ω2

∼= 0[

−1 + sω21 − [l(b+ a)− l]ν21

]

a ∼= 0

−(1− sν21) a− ν21 b− 2bω1τ1 Ω2∼= 0

msτ1√p Ω3 +msω1

√p Ω1

∼= 0

−m

2sω1

√p

(

b+ a

b+ a− l/l+

a

a− l/l

)

−msν1√p Ω3

∼= 0

(103)

The redu ed PDS (95)-(97) an be solved for all the se ond partial derivatives of the unknowns,

namely a, b, l, νa and ωb, if, and only if, the system (103) an be solved for the six unknowns a, b,

l, Ω1, Ω2 and Ω3; that is if, and only if, it has a non-null determinant, ∆ 6= 0, where

∆ := 2bω21ν1τ1 p

[

1− sω21 + [l(b+ a)− l]ν21

] l − al

l2[(

b+ a− l

l

)

(1− sν21)− sω21

(

a− l

l

)]

(104)

whi h stands for the hara teristi determinant of the partial dierential system (95-97).

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